Dec., 3922
T H E JOURNAL OF INDUSTRIAL A N D ENGINEERING CHEMISTRY
1101
Heat Transfer by Conduction and Convection’” 11-Liquids Flowing through Pipes By W.H. McAdamsS and T. H. Frost‘ MASSACHUSETTS INSTITUTE’OB TECHNOLOGY, CAMBRIDGE, MASS.
I n this article a reoiew of the literature has been made, including Reynolds’ analogy between heat transfer and friction. A simplified form of the theoretical Boussinesq equation is proposed for the film coeficieni of heat transfer, h, for liquidflowing inside pipes in turbulent motion. This equation is based on dimensional considerations, and shows the relation between the coeficient, h, and thermal conductioity, diameter, linear uelocity, uiscosity, and density. According to this equation, the coeficient uaries directly as approximately the 0.8 power of the product of linear oelocity and density divided by the uiscosity of the film, directly as the first power of the thermal conductioity, and inuersely as approximately the 0.2 power of the inside diameter of the pipe. This equation has been tested, with experimental data for water from four sources, and with data for light oils from two soumes by
plotting hD/$ versus Dupjz. As shown by Fig. 3. a straight line may be drawn through the experimental points on logarithmic paper. The coeficient may be predicted by use of Fig. 3, or from the equation of the curw:
(For meaning of symbols, see Nomenclature Table.) = 0.329 and p is about 62. this equation For water, since becomes 199 u 0.795 = Djj4(;) The fouling of pipes is discussed, and a factor of safety is suggested.
N A PREVIOUS article1)*on this subject the authors pointed out the advantages of studying heat transfer from the point of view of film coefficients rather than over-all coefficients; this discussion is on the same basis. POSSIBILITY OF T W O TYPES O F FLUID MOTIOX It is well established that a fluid flowing through a pipe may assume either of two types of motion-nameIy, straight line (viscous) motion or turbulent (eddy) motion. A full discussion of this matter is given in a recent paper,2 and it is there shown that, the “critical” velocity-i. e., the average velocity of flow at which straight line motion tends to break over into the more common turbulent motion-is figured by the following equation:
Act. D Scom-The greater part of experimental data for the flow of liquids has been confined to the flow of water through pipes of various diameters. Fortunately, the velocities used in commercial practice are nearly always far above the critical velocity, and this paper will deal primarily with the flow of water under these conditions through tubes or pipes. The possibility of the two types of motion has been mentioned
In this equation uc is the critical velocity in ft. per sec.namely, the cu. ft. per sec. flowing through the pipe divided by its cross sectional area in sq. ft.; p represents the absolute viscosity of the fluid in lbs. per sec. per ft.; d represents the inside diameter of the pipe in feet, and p is the density of the liquid at its average temperature in Ibs. per cu. ft. However, it was there shown that in some cases unstable viscous motion may develop, with the result that the break does not occur until the velocity becomes three times that calculated from the foregoing formula. In other words, the velocity at which one type of motion ceases and the other begins is rather uncertain. Nevertheless, calculations of critical velocity should be made in all cases, inasmuch as equations for the heat-transfer film coefficient for the common type of flow (turbulent motion) may not hold for viscous motion. This is because in turbulent motion the liquid a t any cross section is continually being mixed by eddy currents, the main resistance being found in the relatively stationary film of liquid a t the wall, whereas in viscous motion eddy currents are absent and the heat must flow between the pipe and the
( aLyooo/ 7 . t U 1 J l 3 4 1areca/ i Ill
I
1
Received August 21, 1922.
a Published as Contribution No. 77 from the Department of Chemical
Engineering, Massachusetts Institute of Technology. 8 Associate Professor of Chemical Engineering, Massachusetts Institute of Technology. 4 Instructor of Chemical Engineering, Massachusetts Institute of Technology. Numbers in the text refer to the Bibliography, p. 1104.
*
main body of liquid solely by the process of conduction through a layer which is thicker than in the case of turbulent motion. Fig. 1 shows how the critical velocity varies with the “kinematic” viscosity ( p / p ) for a pipe 0.625 in. in diamother sizes multiply ordinates by
4 ITIlJil,
FIQ.CRITICAL
1111 !
a m!/ !
-*
4 41 71 i l J J l I I o.oo/
4
VELOCITIES IN PIPES
a t this point for two reasons: first, because occasionally the cooling pipes through which the water flows may be so small and the velocity so low that turbulent flow will not be found; and second, because the flow of liquids more Viscous than water will be touched upon in this article.
1102; ---
THE JO URNAI; OF IND WSTRIAL AIfD ENGINEERING CHEMISTRY
Vol. 14, No. 1 2
1800 a c (pun) I*'-m THEORETICAL DISCUSSION k = (71 ," ., d1-?8 REYNOLDS' DERIVATION: ANALOQY BETWEEN HEATTRANSFER AND FRICTION-In 1874 Reynoldsa derived a theoretical where, for convenience, 1 y has been replaced by n. This equation states that the film coefficient of heat transequation for the coe5cient of heat transfer for the case under discussion. This was based on the assumption of a certain fer should be a direct function of the linear velocity, density relation between heat transfer and his law for the relation and viscosity, and an inverse function of diameter. For between the friction factor and a certain group of variables. example, for turbulent motion where n = 0.8 (and. y = The latter has been given a thorough test and found t o be -0.2), this becomes 1800 a C ( p ~ ) ' J . * p ~ . ~ valid. His derivation is essentially as follows: h= dO.2 (7a) ~
+
Consider a differential length, d x ft., of a pipe of inside radius Tt. through which water at the temperature t o F. is flowing at a rate of w lbs. per sec. and at an average velocity of u f t . per sec. The temperature of the inner wall of the pipe a t this section is "OF., the specific heat of the water is C B. t. u. per Ib. perOF., and the density i s p Ibs. per cu. f t . 1-By means of a heat balance, the rate of heat transfer per unit area of inner pipe wall (Q/A0), expressed as B. t. u. per hr. per sq. ft., may be equated to the heat picked up by the water. Q 3600 w C d t (2) A0= 2 n r d x 2-According to the familiar Fanning equation for friction, the loss of intensity of pressure is Y
(3)
where f is the experimentally determined friction factor and g equals the acceleration due to gravity, 32.2 f t . per sec. per see_. 3-Assume that the pressure lost due to friction ( n r 2 d p ) divided by the momentum of the water (wulg) equals the heat transferred to the water divided by the heat which would have been transferred if the water had been warmed to the temperature of the wall. w2db w C dt ,", -= (*J w C (T - t ) wu/i By combining these equations one obtains
In 1897 Stanton6 tested this relation with his data for water flowing through 18-in. lengths of several small sizes (0.29, 0.42, and 0.55 in.) of copper tubes. The value of n for Equation 7 was found to be about 0.83, and Reynolds' equation allowed satisfactorily for variations in h due to velocity and diameter, but did not allow for the actual efect of viscosity. Stanton found empirically that Equation 7a divided by the first power of the viscosity would fit his datanamely, where a1 is a new constant. However, Stanton abandoned Equation 7b, apparently because it was different from Reynolds' equation, and adopted the following empirical equation:
where az, a,and p are new constants.
(5)
14 should be noted that the last equation is Newton's law solved for the film coefficient of heat transfer, h, between the pipe wall and the water. A careful inspection of Equation 3 shows that the friction factor has no dimensions. In 1883 Reynolds4 predicted that f should be some function of certain variables grouped so as to have no dimensions. It was known that four variables affected f-namely, diameter, d , velocity, u, density, p , and absolute viscosity, p. Noting that the dimensions of absolute viscosity in English absolute units are lbs. per sec. per ft., it is seen that the arrangement dUp EI Cft.) (ft./sec.) (lbs./cu. ft.) (6) P (lbs.)/(sec.) (ft.) has no dime?isdons. Furthermore, it is the only one filling this requirement. Reynolds assumed the relation between f and d u p l p to be an exponential one, and this has been found from many experiments4 to be the case. For turbulent motion f is found to vary as some exponent y, which changes very slowly with large variations in d u p l p . The relation between f and d u p / p is shown in Fig. 2. The right-hand branch of the curve is seen to be quite flat, although the slope is changing very slowly. For this branch, f = a ( d u p j p ) ~ an , average line would give f = a(dup/p)-0.2 (60) The left-hand branch, for viscous motion, has the equation f = 16(dup/p)-' (Ob) the value of y being constant.
These predictions of Reynolds have greatly simplified calculations for the flow of liquids and gases through pipes, as values of f for all conditions may be read as ordinates from a single plot (such as Fig. 2 ) with d u p / p as abscissae. Practical application of this relation has been made elsewhere, bfit it was introduced here for reasons which appear below. ,Instead of usingf in Equation 5 as a variable, Reynolds substituted its equivalent from Equation 6a, obtaining for turbulent motion:
OI
F I O . .%-VARIATION
OF
FRICTION FACTOR (f)IN
EQUATION
3 WITH
dup P
DERIVATION OF RATIONAL FORMULAS-It seems strange t h a t Reynolds did not apply the same process of "dimensional reasoning" to the heat transfer problem, which proved so valuable for the pressure drop relations. By definition, the film coefficient h equals the thermal conductivity k divided by the effective film thickness L-i. e., h = K/L. Values of k may be easily obtained from tables of physical constants,? so the only problem is the prediction of what variables determine L. L has linear dimensions. While both length and inside diameter fill this requirement, it does notseem reasonable that the length of the pipe has any effect on the coefficient. Since there are indications6 that the zone of relatively low velocity in a pipe-referred to as the film of effective thickness, L-is directly proportional to the diameter of the pipe, other things being equal, it will be assumed that L varies as d. The effective thickness of film is known to be a function of several variables, such as velocity and viscosity. Since the friction factor and the
__
' t See also Reference 1 for means of predicting k by H. F. Weber's equation.
THE JOURNAL OF INDUXTRIAL k?\rD ENGINEERING CHEMIXTRY
Dec., 1922.
1103
FILMCOBFFICIENTFOR LIQUIDS(hL) FLOWING TEROUQH PIPESIN TURBULENT MOTION
*-22.6 R DUP , .-D( y ) , where h, B. t. u. per hr. per ft. of film area per 0.79'
=
sq.
O
F. drop,
pipe to liquid.
K = Thermal Conductivity of stationary liquid, B. t. u. D
u
5
per hr. per sq. f t . per OF. per f t . Actual i. d. of pipe in inches. Av. velocity of liquid in f t . per sec. = Av. density of the liquid in. lbs. per cu. f t . = Av. viscosity of liquid /Zm relative t o water a t 138' F. =
I=
FIG.3
film thickness are determined by the degree of turbulence (dup/p), it seems reasonable to suppose that this same "dimensionless" ratio would determine the effective film thickness for heat transfer. Hence, we assume (9)
where b and n are to be experimentally determined. It should be noted that this equation is a logical one, in that all dimensions cancel. Rearranging,
in lbs./sec./ft., p , has been replaced by relative viscosity in centipoises, 2. Since water at 68" F. (20" C . ) has a viscosity of 1.00 centipoise, the viscosity in centipoises is numerically the same as viscosity relative to water a t 68" F., a concept which may be readily visualized, (It should be noted that z is the reciprocal of the relative fluidity f used in our preceding article.) Hence, we shall plot hD/k versus Dup/z to determine the value of b and n in the expression
I n 1909, Nusselt8 applied the "principle of physical homogeneity" to the problem of predicting the relation between the film coefficient of heat transfer for gases and certain variables. As a result he obtained the following equation:
This checks with Stanton's experiments on viscosity. However, he apparently abandoned it and adopted Equation h = ark dUp 8, which contaihs the two empirical correction factors for temperature. (In 1912, Stanton' derived Equation 9a He then assumed n = m, giving for gases from Equation 7a by assuming that C = a&/p, neglecting variations in y , the ratio of specific heat a t constant pressure to that at constant volume, and assuming US to be substantially constant.) which is the same as the Boussinesqg equation derived in Equation Sa indicates that the film coefficent h for a 1905. (However, Boussinesq did not test this theoretical given sized pipe and fluid varies directly as the product of equation with data in order t o prove its validity and to the linear velocity and the density to the power n, and determine the constants necessary for its use.) Nusselt inversely as the viscosity to the same power. Furthermore, used Equation 10a in correlating his data for various gases the slope of the curve or value of the exponent n theoretically flowing inside pipes. This equation fitted the data very may be predicted from the friction factor plot, since 1 y well, but mainly because of the use of the term (up)". Other = n. Equation 9 indicates the general relation between equations, employing (up)", but containing terms differept the film coefficient h and linear velocity, density, viscosity, from those in Equation loa, also fit the dafa. diameter, and thermal conductivity. It is obvious that It should be noted that Equation 9 may be obtained from the logical method of testing this theoretical relation is to the Boussinesq equation by placing m = 0. Insfead of plot hd/k versus d u p / p on logarithmic paper. If the theory m being equal to n, or m being equal to zero, it may be that is correct, the slope will be practically constant over a con- both m and n are finite,' which would require a three-casiderable range of dup/p, and b may be determined from the ordinate or "space" diagram instead of the one employed ordinate where d u p / p equals unity. in Fig. 3. If this is true, the coordinates would be (hd/k), As a matter of convenience, diameter in feet, d, has been (dup/p), and ( C p / k ) . It is possible that the Boussinesq replaced by diameter in inches, D , and absolute viscosity equation may prove neral equation for
a( )" (9)" ~
+
T H B JOURNAL OF INDUSTRIAL AND ENGINEERING CHEMIXTRY
1104
the film coefficient of heat transfer for all fluids flowing in pipes. I n recent years Buckingham’O has published papers dealing with dimensional equations for heat transfer for fluids flowing in pipes. While his equations were of the same genera nature as those discussed above, they are not identical and apparently were not tested with experimental data.
EXPERIMENTAL DATA WATER-Considerable data have been collected in these laboratories to determine the film coefficient of heat transfer for water flowing through pipes. The apparatus, which was similar to that described in our previous article, permitted the simultaneous determination of the coefficients, both on the water side and on the steam side. The coefficients were calculated from the data as previously described. I n addition to these data collected by Wishnew” and Trowbridge12 in connection with undergraduate theses, data for water from two other sources13 have.been calculated to our basis and are also shown in Fig. 3. The range of variables in Fig. 3 is as follows: h
................. 2700 160 ..................
Lowest.. Highest
,
U
0.6 20.0
D #/a
2
e 0.8
Dud2 25 1800
1.0
OIL-On this same plot are included a few data for mineral oils of low viscosity. While these datal4 are not so reliable as those for water, owing to the lack of exact information concerning certain physical properties, ‘yet they are compatible with them. Hence, in the lack of additional data for oil, the use of Fig. 3 is recommended for purposes of estimation. Below the critical velocity y = -1, and since y 1 = n, n should be equal to zero for viscous motion. Under these conditions Equation 9b reduces to bk
+
h=-jj-
No data are a t present available to determine whether Equation 9c will hold under these conditions. Hence, the use of the plot below the critical velocity is not recommended. As shown by Equation 1, d u p / p = 16/j0 = 942, a t the theoretical critical velocity, which is equivalent to an abscissa in Fig. 3 of Dup/z of about 7.6.t If unstable viscous motion develops, Dup/z might be as high as 23. However, the probability of unstable viscous motion where heat is being transferred is remote. Hence, the curve in Fig. 3 should not be used below an abscissa of 8. FOULING OF PIPES-It has long been known that in heattransfer apparatus the inner surfaces of the pipes carrying certain liquids become covered with a deposit of solid matter. For example, where the cooling water contains dissolved bicarbonates such as calcium, these decompose under the action of heat and a scale of carbonate is formed. Further, a slime is often deposited, due to dirt in the cooling water. In the case of oils, deposits containing naphthalene, paraffin, and other materials may form. The scale or deposit in any case adds another resistance to heat transfer, which varies directly with the thickness and nature of the deposit. Allowance made for this should be by adding another resistance term to the denominator of Equation 1 of our previous article. Instead, it is customary to multiply the liquid film coefficient by a cleanliness factor, ct, which never exceeds unity. The use of such a cleanliness factor is not theoretically sound, as in a given pipe containing a definite scale, the value of c1 might be 0.4 with a very high liquid film coefficient and 0.8 with a very low coefficient. t For smooth pipes fc is about 88.3 per cent that in rough pipes, hence these values would be increased about 13 per cent.
Vol. 14,No. 12
I n 1921, Frost and Manley15 found the film coefficient in 2-in. standard steel pipe after two months’ use with water decreased to 75 per cent of its value in the new pipe-i. e., c1 = 0.75. After a thorough cleaning the original values were obtained. When very bad cooling water is encountered, or where napthalene, paraffin, etc., may deposit from oils, a value of c1 = 0.5 is suggested. OPTIMTJU VELOCITY OF LIQUIDIN
A
CONDENSER
With data available for the film coefficient from condensing vapor to solid, it is possible to calculate the over-all coefficient in apparatus such as condensers, water heaters, etc., for any given conditions. Knowing the costs of heating surface and pumping,S one can make an economic balance to determine the optimum liquid velocity. The method of calculation has been indicated in a recent paper’n for the case of the heating of air by steam. NOMENCLATURE TABLE English Symbols a (ai,as, as) = Constants. b = Constant
C d D
f g
9
k(K)
L m n P
Q r
T t
u W 5
Y 2
a
B Y T
P P
= Specific heat of liquid = B. t. u. /lb./’F. = caI./g./OC. = Actual inside diameter of pipe in feet. Actual inside diameter of pipe in inches. = Friction factor (no units). = Acceleration due t o gravity 32.2 ft./sec./sec. = Coefficient of heat transfer through liquid film inside of pipe = B. t. u/hr./sq. f t . of inner wall of pipe per O F. difference in temperature between inner wall of pipe and liquid a t the center line of pipe. = Thermal conductivity of stationary liquid = B. t. u./ hr./ sq. ft./ F. per foot of thickness. = Effectivethickness of film, in feet. = Exponent. 0 Exponent = 1 y. = Intensity of absolute pressure, lbs./sq. f t . of cross section. = Quantity of heat transferred a t right angles t o heat transfer surface, B. t. u. 0 Inside radius of pipe in f t . = d/2. = Temperature of inner wall of pipe, O F. = Temperature of water a t center line of pipe, O F. = Average water velocity, ft./sec. Average rate of flow of water, lbs./sec. = Length of pipe in f t . = Exponent a t average film temperature. = Viscosity of liquid a t average film temperature relative to water a t 6S0 F. (20° C.) as unity = centipoises.
-
-
+
-
Greek Symbolp = Constant. = Constant. = Ratio of specific heat of gas a t constant pressure to that at constant volume. 3.14. = Density of liquid a t center line temperature, lbs./cu. f t . = Absolute viscosity of liquid, lhs./sec./ft , = 0.0672 poises = 0.000672 centipoises.
-
BIBLIOGRAPHY 1-McAdams and Frost, J . Ind. Eng. Chem., 14 (1922). 13. 2-Wilson, McAdams, and Selzer, J. Ind. E n g . Chem., 14 (1922), 105. 3-Reynolds, Proc. Manchester Lit. and Phil. Soc., 1874, 8 ; see also Reference 4. 4-Reynolds, Trans. Roy. SOC. London, 1888, 158. See also Stokes, Math. and Phys. Pa$ers, 8 (1850), 17; Helmholtz, Wissenschajtlichle Abhandlungen, 1 (1873), 158; Rayleigh, Phil. Mag., 48 (1889), 321; Lamb, “Hydrodynamics of Fluids,” Cambridge University Press, 4th ed., p. 653. 5-Stanton, Phil. Trans., lSOA (1897), 67. &Stanton, “The Mechanical Viscosity of Fluids,” Proc. Roy. SOC. London, 8611 (1911), 366; “Collected Researches,” National Physical Laboratory, 8 (1912), 73.
8 Accurate data are now available for the calculation of the Powrequired to pump fluids through pipes made of various materials. See Reference 2.
